1. Heat milk with sugar and salt in small saucepan just to lukewarm.
2. Sprinkle yeast into very warm water in a large bowl. ("Very warm water" should feel comfortably warm when dropped on wrist.) Stir until yeast dissolves, then stir in cooled milk mixture.
3. Beat in 5 cups flour and 1 cup cheese until completely blended. Beat in remaining flour gradually to make a soft dough.
4. Turn out onto lightly floured surface; knead until smooth and elastic, adding only enough flour to keep dough from sticking.
5. Place in greased large bowl; turn to coat all over with shortening; cover with a clean towel. Let rise in warm place, away from drafts, 1 hour, or until double in bulk. Stir garlic into butter. Brush pans or casseroles with garlic butter.
6. Punch dough down; knead 1 minute on lightly floured surface, then shape this way: For large loaves, divide dough in half, divide each half in 7 even pieces, shape into rolls; place 6 rolls around edge of prepared pan and 1 in center. For medium-size loaves: Divide dough into 16 even pieces; shape into rolls, place 2 rolls in each of 8 prepared ten-ounce casseroles or custard cups. For miniature loaves: Divide dough into 14 pieces; shape into loaves; place in prepared toy-size loaf pans; cover. Let rise again in warm place, away from drafts, 45 minutes, or until double in bulk. Brash tops with water; sprinkle lightly with extra Parmesan cheese.
7. Bake in very hot oven (400°) 40 minutes for large loaves, 30 to 35 minutes for small and medium loaves, or until breads sound hollow when tapped. Remove from pans to wire racks; cool completely.